1. **State the problem:** Solve the equation $3x^2 + 9x = -6$ by graphing the related function and finding its zeros. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero: $$3x^2 + 9x + 6 = 0$$ 3. **Define the function:** Let $$y = 3x^2 + 9x + 6$$ 4. **Find zeros of the function:** The zeros are the values of $x$ where $y=0$. 5. **Use the quadratic formula:** For $ax^2 + bx + c = 0$, the solutions are $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=3$, $b=9$, $c=6$. 6. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 9^2 - 4 \times 3 \times 6 = 81 - 72 = 9$$ 7. **Calculate the roots:** $$x = \frac{-9 \pm \sqrt{9}}{2 \times 3} = \frac{-9 \pm 3}{6}$$ 8. **Find each root:** - For $+$: $$x = \frac{-9 + 3}{6} = \frac{-6}{6} = -1$$ - For $-$: $$x = \frac{-9 - 3}{6} = \frac{-12}{6} = -2$$ 9. **Interpretation:** The graph of $y = 3x^2 + 9x + 6$ crosses the x-axis at $x = -1$ and $x = -2$. These are the solutions to the original equation. **Final answer:** $$x = -1, -2$$